Let $X$ be a second-countable Hausdorff smooth manifold, let $\pi:E \to X$ be a smooth complex vector bundle of finite complex rank $r \in \mathbb{N}$, and let $h$ be a smooth Hermitian metric on $E$, with the convention that $h$ is complex-linear in the first argument and conjugate-linear in the second. A complex-linear connection on $E$ means an operator
for every $V \in \mathfrak{X}(X)$, $s \in \Gamma(E)$, and $f \in C^\infty(X;\mathbb{C})$. Then there exists a complex-linear connection $\nabla$ on $E$ that is compatible with $h$, meaning that for every smooth real vector field $V \in \mathfrak{X}(X)$ and every pair of smooth sections $s,t \in \Gamma(E)$,
Moreover, if $\nabla_0$ is one metric-compatible complex-linear connection on $(E,h)$, then the set of all metric-compatible complex-linear connections on $(E,h)$ is the affine space
where $\mathfrak{u}(E,h) \subset \operatorname{End}(E)$ is the smooth real vector subbundle whose fibre at $x \in X$ consists of all complex-linear endomorphisms $A:E_x \to E_x$ satisfying
paragraph
admin
\begin{align*}
h_x(Au,v)+h_x(u,Av)=0
\end{align*}
latex_env
admin
for all $u,v \in E_x$, and where $\Omega^1(X;\mathfrak{u}(E,h))$ denotes the space of smooth real one-forms on $X$ with values in this subbundle. Equivalently, every metric-compatible connection $\nabla$ is uniquely of the form